A light plane pilot reported a NMAC with another light aircraft during climb. Pilot reported event to tower and flight continued.

Narrative

Received an IFR departure clearance from runway 23 with instructions from tower controller to turn heading 290 and climb and maintain 3000 on departure. Upon reaching safe altitude; gear raised and aircraft in climb configuration; began a right turn and stabilized in climb on heading 290. Aircraft was climbing into the sun. Right as tower controller called to handoff to Potomac approach; I observed Aircraft Y passing opposite direction; off my right wing; at the same altitude; probably less than 4-5 wingspans separation (<200 ft). Course ADS-B track data shows no greater than 400 feet of separation. I immediately responded to the tower controller; 'I need to report a near miss; Aircraft Y off my right wing; less than 300 feet'. Tower controller apologized and stated that the other aircraft was turned inbound on a visual approach and that he had failed to advise of the traffic. I never saw the aircraft due to being turned into the sun and the other aircraft clearly did not see me either. My aircraft was fully illuminated at the time (LDG lights; anticollision; beacon; position). This incident was nearly catastrophic.This occurred during the ATC furlough and the tower controller at FDK was simultaneously working ground; tower; and clearance delivery tasks.